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Advanced Mathematics MSc FM, MTF, QF Induction Programme Advanced Mathematics Academic year 2010/2011 Lecture notes prepared by Laura Ballotta and George Harrison Section I Linear Algebra Differential Calculus Integral Calculus Difference and Differential Equations Table Content 1 LINEAR ALGEBRA .................................................................................................................................... 5 1.1. WHAT IS A MATRIX?................................................................................................................................. 5 1.2. SPECIAL TYPES OF MATRICES .................................................................................................................... 6 1.2.1. Vectors............................................................................................................................................ 6 1.2.2. Square matrix.................................................................................................................................. 6 1.2.3. Diagonal matrix .............................................................................................................................. 7 1.2.4. Identity Matrix................................................................................................................................ 7 1.3. MATRIX OPERATIONS ............................................................................................................................... 7 1.3.1 Adding and subtracting matrices .................................................................................................... 7 1.3.2 Scalar multiplication....................................................................................................................... 8 1.3.3. Transpose of a Matrix..................................................................................................................... 8 1.3.4. Vector multiplication ...................................................................................................................... 9 1.3.5 Matrix multiplication .................................................................................................................... 10 1.3.6 Properties of matrix multiplication ............................................................................................... 12 1.4. THE DETERMINANT................................................................................................................................. 13 1.4.1. How to find the determinant of a (2 x 2) matrix ........................................................................... 13 1.4.2. The determinant of a (3 x 3) matrix .............................................................................................. 14 1.4.3. Method of cofactors in determining determinants ........................................................................ 14 1.4.4. Properties of Determinants ........................................................................................................... 17 1.5. INVERSE OF A MATRIX............................................................................................................................. 18 1.5.1. Definition of an inverse matrix..................................................................................................... 18 1.5.2. How to find the inverse of a square matrix................................................................................... 18 1.6. TRACE OF A MATRIX............................................................................................................................... 19 1.7. SYSTEM OF LINEAR EQUATIONS ............................................................................................................. 20 1.8. PARTITIONED MATRICES......................................................................................................................... 22 1.9. VECTOR SPACES ...................................................................................................................................... 22 1.9.1 Rank of a matrix ........................................................................................................................... 25 1.10 EIGENVALUES AND EIGENVECTORS ................................................................................................... 26 1.10.1 General results for characteristic roots and vectors ...................................................................... 28 1.10.2 Further results on determining the rank of a matrix.................................................................... 29 1.11 DEFINITE MATRICES .......................................................................................................................... 29 2. DIFFERENTIAL CALCULUS .................................................................................................................. 32 2.1. DIFFERENTIATION ................................................................................................................................... 32 2.1.1. Definitions .................................................................................................................................... 32 2.1.2. Rules of Differentiation ................................................................................................................ 35 2.1.3. Derivatives of Exponential and logarithmic functions.................................................................. 37 2.1.4. Higher order derivatives............................................................................................................... 38 2.2. APPLICATIONS OF DERIVATIVES ............................................................................................................. 39 2.2.1. Taylor and Maclaurin Series......................................................................................................... 39 2.2.2. Increasing and decreasing functions ............................................................................................. 41 2.2.3. Concavity and Convexity.............................................................................................................. 42 2.2.4. First derivative test for relative extremum.................................................................................... 44 2.2.5. Second derivative test for relative extremum................................................................................ 45 2.2.6. The Nth-derivative test................................................................................................................. 46 2.2.7. Economic applications of derivatives ........................................................................................... 46 3. DIFFERENTIAL CALCULUS (THE MULTIVARIATE CASE) ............................................................. 53 3.1. MULTIVARIATE FUNCTIONS .................................................................................................................... 53 3.2 LIMITS AND CONTINUITY......................................................................................................................... 53 3.3 DIFFERENTIATION ................................................................................................................................... 55 3.4 APPLICATION OF DIFFERENTIATION ........................................................................................................ 61 3.4.1 Critical Values of Bivariate Functions.......................................................................................... 61 3.4.2. Critical Points of Multivariate Functions...................................................................................... 64 3.4.3. Constrained Optimisation ............................................................................................................. 67 4 INTEGRAL CALCULUS........................................................................................................................... 72 4.1. INTEGRATION.......................................................................................................................................... 72 2 4.1.1. Definite and Indefinite Integrals ................................................................................................... 72 4.1.2. Properties of definite integrals...................................................................................................... 74 4.1.3. Rules of Integration - Indefinite Integral ...................................................................................... 75 4.1.4. Auxiliary Conditions..................................................................................................................... 79 4.2. APPLICATIONS OF INTEGRALS ................................................................................................................. 79 4.2.1. Deriving Totals (Revenue/Cost) from Marginals (Revenue/Cost)................................................ 79 4.2.2. Present Value of Cashflows.......................................................................................................... 80 4.2.3. Measuring Probabilities ................................................................................................................ 81 4.3 MULTIPLE INTEGRATION......................................................................................................................... 81 4.3.1. Double Integral of a Function of Two Variables ..........................................................................81
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